Geodesic
Functional Integration with an “Automorphic” Boundary Condition and Correlators of Third Components of Spins in the $XX$ Heisenberg Model
Teoretičeskaâ i matematičeskaâ fizika, Tome 136 (2003) no. 2, pp. 285-298 Cet article a éte moissonné depuis la source Math-Net.Ru

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For the generating function of static correlators of the third components of spins in the $XX$ Heisenberg model, we derive a new representation given by a combination of Gaussian functional integrals over anticommuting variables. A peculiarity of the resulting functional integral is that a part of the integration variables depend on the imaginary time automorphically: these variables are multiplied by a certain complex number under the shift of the imaginary time by the period. The other variables satisfy the standard boundary conditions of the fermionic/bosonic type. Functional integration results are represented as determinants of matrix operators. We finally evaluate the generating function of correlators and the partition function of the model in the zeta-function regularization. The consistency of the suggested functional definition is confirmed by calculating several correlation functions of the third components of spins at a nonzero temperature.
Keywords: functional integration, $XX$ Heisenberg model, correlators, generalized zeta function. 
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     author = {K. L. Malyshev},
     title = {Functional {Integration} with an {{\textquotedblleft}Automorphic{\textquotedblright}} {Boundary} {Condition} and {Correlators} of {Third} {Components} of {Spins} in the $XX$ {Heisenberg} {Model}},
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K. L. Malyshev. Functional Integration with an “Automorphic” Boundary Condition and Correlators of Third Components of Spins in the $XX$ Heisenberg Model. Teoretičeskaâ i matematičeskaâ fizika, Tome 136 (2003) no. 2, pp. 285-298. http://geodesic.mathdoc.fr/item/TMF_2003_136_2_a7/

[1] N. M. Bogolyubov, A. G. Izergin, V. E. Korepin, Korrelyatsionnye funktsii integriruemykh sistem i kvantovyi metod obratnoi zadachi, Nauka, M., 1992 | MR | Zbl

[2] A. Lenard, J. Math. Phys., 5:7 (1964), 930–943 ; 7:7 (1966), 1268–1272 | DOI | MR | DOI | MR

[3] V. E. Korepin, N. A. Slavnov, Commun. Math. Phys., 129:1 (1990), 103–113 | DOI | MR | Zbl

[4] F. Colomo, A. G. Izergin, V. E. Korepin, V. Tognetti, Phys. Lett. A, 169:4 (1992), 243–247 ; ТМФ, 94:1 (1993), 19–51 | DOI | MR | MR

[5] A. G. Izergin, A. R. Its, V. E. Korepin, N. A. Slavnov, Algebra i analiz, 6:2 (1994), 138–151 | MR | Zbl

[6] A. A. Slavnov, L. D. Faddeev, Vvedenie v kvantovuyu teoriyu kalibrovochnykh polei, Nauka, M., 1988 | MR | Zbl

[7] V. N. Popov, Kontinualnye integraly v kvantovoi teorii polya i statisticheskoi fizike, Atomizdat, M., 1976 | MR

[8] V. S. Kapitonov, K. N. Ilinskii, Zap. nauchn. semin. POMI, 224, 1995, 192–207

[9] A. G. Izergin, V. S. Kapitonov, N. A. Kitanin, Zap. nauchn. semin. POMI, 245, 1997, 173–206 | Zbl

[10] V. S. Kapitonov, A. G. Pronko, Zap. nauchn. semin. POMI, 269, 2000, 216–261 | MR

[11] F. A. Berezin, Metod vtorichnogo kvantovaniya, Nauka, M., 1986 | MR | Zbl

[12] A. G. Izergin, N. A. Kitanin, N. A. Slavnov, Zap. nauchn. semin. POMI, 224, 1995, 178–191

[13] U. Bilstein, B. Wehefritz, J. Phys. A, 32:2 (1999), 191–233 ; U. Bilstein, V. Rittenberg, J. Phys. A, 33:42 (2000), L381–L385 | DOI | MR | Zbl | DOI | MR | Zbl

[14] V. Korepin, J. Terilla, Quantum Information Processing, 1 (2002), 225–242 ; E-print quant-ph/0202054 | DOI | MR

[15] N. Kitanine, J. M. Maillet, N. A. Slavnov, V. Terras, Nucl. Phys. B, 642 (2002), 433–455 ; E-print hep-th/0203169 | DOI | MR | Zbl

[16] T. Kojima, V. Korepin, N. Slavnov, Commun. Math. Phys., 188:3 (1997), 657–689 ; 189:3, 709–728 ; V. E. Korepin, N. A. Slavnov, J. Phys. A, 31:46 (1998), 9283–9295 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[17] L. Alvarez-Gaumé, “Supersymmetry and index theory”, Supersymmetry, NATO ASI Ser., 125, eds. K. Dietz et al., Plenum, New York, 1985, 1–44 | DOI | MR

[18] K. Malyshev, Zap. nauchn. semin. POMI, 269, 2000, 269–291 ; C. Malyshev, XX Heisenberg spin chain and an example of path integral with “automorphic” boundary conditions, E-print hep-th/0204007 | Zbl

[19] S. W. Hawking, Commun. Math. Phys., 55:2 (1977), 133–148 ; J. S. Dowker, R. Critchley, Phys. Rev. D, 13:12 (1976), 3224–3232 | DOI | MR | Zbl | DOI | MR

[20] S. Khoking, “Integraly po traektoriyam v prilozhenii k kvantovoi gravitatsii”, Obschaya teoriya otnositelnosti, eds. S. Khoking, V. Izrael, Mir, M., 1983, 363–406

[21] D. B. Ray, I. M. Singer, Adv. Math., 7:2 (1971), 145–210 | DOI | MR | Zbl

[22] E. Corrigan, P. Goddard, H. Osborn, S. Templeton, Nucl. Phys. B, 159:3 (1979), 469–496 | DOI | MR

[23] A. S. Schwarz, Commun. Math. Phys., 64:3 (1979), 233–268 | DOI | MR | Zbl

[24] B. P. Dolan, Ch. Nash, Commun. Math. Phys., 148:1 (1992), 139–153 ; R. Kantowski, K. A. Milton, Phys. Rev. D, 36:12 (1987), 3712–3721 | DOI | MR | Zbl | DOI | MR

[25] V. N. Romanov, A. S. Shvarts, TMF, 41:2 (1979), 190–204 | MR

[26] A. S. Shvarts, Kvantovaya teoriya polya i topologiya, Nauka, M., 1989 | MR

[27] E. Lieb, T. Schultz, D. Mattis, Ann. Phys., 16:3 (1961), 407–466 ; Th. Niemeijer, Physica, 36:3 (1967), 377–419 ; 39:3 (1968), 313–326 | DOI | MR | Zbl | DOI | DOI

[28] W. L. Baily, Jr., Introductory Lectures on Automorphic Forms, Princeton Univ. Press, Princeton, NJ, 1973 ; И. Р. Шафаревич, Основания алгебраической геометрии, Наука, М., 1988 | MR | Zbl | MR

[29] V. N. Popov, S. A. Fedotov, ZhETF, 94:3 (1988), 183–194 | MR

[30] V. A. Fok, Izv. AN SSSR. Ser. fiz., 1937, no. 4–5, 551–568; J. Schwinger, Phys. Rev., 82:5 (1951), 664–679 | DOI | MR | Zbl

[31] W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer, Berlin, 1966

[32] A. S. Schwarz, Lett. Math. Phys., 2:3 (1978), 247–252 | DOI | MR | Zbl

[33] F. R. Gantmakher, Teoriya matrits, Nauka, M., 1988 | MR | Zbl